Pair of Opposite Sides of a Parallelogram are Equal and Parallel
Here we will discuss about one of the important geometrical property of parallelogram.
A quadrilateral is a parallelogram if one pair of opposite sides are equal and parallel
Given: PQRS is a quadrilateral in which PQ = SR and PQ ∥ SR.
To prove: PQRS is a parallelogram.
Construction: Join PR and QS such that they intersect at O.
Proof:
|
Statement |
Reason |
|
In ∆OPQ and ∆ORS, 1. ∠OPQ = ∠ORS |
1. PQ ∥ SR and PR is a transversal. |
|
2. ∠POQ = ∠ROS |
2. Opposite angles are equal. |
|
3. PQ = RS |
3. Given. |
|
4. ∆OPQ ≅ ∆ORS Therefore, OP = OR, OQ = OS. In ∆OPS and ∆OQR, |
4. By AAS criterion of congruency. CPCTC |
|
5. OP = OC, OQ = OS, ∠POS = ∠QOR |
5. By statement 4 and reason 2. |
|
6. ∆OPS ≅ ∆OQR Therefore, PS = QR, ∠OPS= ∠ORQ |
6. By SAS criterion of congruency. CPCTC |
|
7. PS ∥QR. |
7. Alternate angles are equal. |
|
8. PQRS is a parallelogram (Proved). |
8. PQ ∥ SR and statement 7. |
Corollary: In a parallelogram, each pair of opposite sides are parallel as well as equal.
From Pair of Opposite Sides of a Parallelogram are Equal and Parallel to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Recent Articles
-
Perimeter of a Square | How to Find the Perimeter of Square? |Examples
Apr 25, 24 05:34 PM
We will discuss here how to find the perimeter of a square. Perimeter of a square is the total length (distance) of the boundary of a square. We know that all the sides of a square are equal. Perimete… -
Perimeter of a Triangle | Perimeter of a Triangle Formula | Examples
Apr 25, 24 05:13 PM
We will discuss here how to find the perimeter of a triangle. We know perimeter of a triangle is the total length (distance) of the boundary of a triangle. Perimeter of a triangle is the sum of length… -
Perimeter of a Rectangle | How to Find the Perimeter of a Rectangle?
Apr 25, 24 03:45 PM
We will discuss here how to find the perimeter of a rectangle. We know perimeter of a rectangle is the total length (distance) of the boundary of a rectangle. ABCD is a rectangle. We know that the opp… -
Dividing 3-Digit by 1-Digit Number | Long Division |Worksheet Answer
Apr 24, 24 03:46 PM
Dividing 3-Digit by 1-Digit Numbers are discussed here step-by-step. How to divide 3-digit numbers by single-digit numbers? Let us follow the examples to learn to divide 3-digit number by one-digit nu… -
Symmetrical Shapes | One, Two, Three, Four & Many-line Symmetry
Apr 24, 24 03:45 PM
Symmetrical shapes are discussed here in this topic. Any object or shape which can be cut in two equal halves in such a way that both the parts are exactly the same is called symmetrical. The line whi…